It has been known that the operator \(\mathcal {A}^\mathcal {S}_{\alpha ,\nu }\)  dominate some classes of classical operators [1]}. For instance, for  \(\nu =1\)   and  \(\nu =2\)   with  \(\alpha =0\)  , these operators dominate large classes of Calderón–Zygmund singular integrals and Littlewood–Paley square functions, respectively. For  \(\nu =1\)   with  \(0<\alpha <d\)  ,  \(\mathcal {A}^\mathcal {S}_{\alpha ,\nu }\)   dominate the fractional integral operator \(I_\alpha \) . However in this note we are unable to prove the result for all the fractional sparse operators. We require the dyadic operators to be even more sparse which we use the following definition.
